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Chapter 4: Bonding in Solids and Electronic Properties

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Chapter 4: Bonding in Solids and Electronic Properties Chapter 4: Bonding in Solids and Electronic Properties Free electron theory Consider free electrons in a metal – an electron gas. •regards a metal as a box in which electrons are free to move. •assumes nuclei stay fixed on their lattice sites surrounded by core electrons, while the valence electrons move freely through the solid. •Ignoring the core electrons, one can treat the outer electrons with a quantum mechanical description. •Taking just one electron, the problem is reduced to a particle in a box. Electron is confined to a line of length a. Schrodinger equation 2 2 2 d d 2me E 2 (E V ) 2 2 2me dx dx Electron is not allowed outside the box, so the potential is ∞ outside the box. Energy is quantized, with quantum numbers n. 2 2 2 n2h2 h2 n n n In 1d: E In 3d: E a b c 2 8m a2 b2 c2 8mea e 1 2 2 2 2 Each set of quantum numbers n , n , h n n n a b a b c E 2 2 2 and nc will give rise to an energy level. 8me a b c •In three dimensions, there are multiple combination of energy levels that will give the same energy, whereas in one-dimension n and –n are equal in energy. 2 2 2 2 2 2 na/a nb/b nc/c na /a + nb /b + nc /c 6 6 6 108 The number of states with the 2 2 10 108 same energy is known as the degeneracy. 2 10 2 108 10 2 2 108 When dealing with a crystal with ~1020 atoms, it becomes difficult to work out all the possible combinations. •A quantity called the wave vector is used and assumed to be continuous. 2 2 2 2 •Substitute kx, ky, kz for nap/a, nbp/b, and ncp/c.E (kx k y kz ) / 2me •kx, ky, kz are considered as components of a vector, k. •k is the wave vector and is related to the momentum of the electron wave. Compare to E = p2/2m, where p is the momentum and m is the mass. All of the combinations of the quantum numbers giving rise to one particular energy correspond to a wave vector of the same length |k|. •These possible combinations for a specific energy produce vectors whose ends lie on the surface of a sphere of radius |k|. •The total number of wave vectors with energies up to and including that with the given energy is given by the volume of the sphere 4pk3/3, where |k| is written as k. •To convert this to the number of states with energies up to the given energy, use the relationships between the components of k and the quantum numbers na, nb, and nc. •Need to multiply the volume by abc/p3. •In order to determine the number of states with a particular energy, define the number of states in a narrow range of k values, dk. •The number of states up to and including those of the wave vector length k + dk is 4/3p2V(k+dk)3 where V (= abc) is the volume of the crystal. •The number with values between k and k + dk is 4/3p2V[(k+dk)3-k3], leading to 4/p2 V k2dk and this quantity is known as the density of states, N(k)dk. •In terms of energy, the DOS is given by: 3 2 3 (2me ) E (V / 2π h )dE 2 Density of States based on the free electron model. -note how the density of states increases with increasing energy (there are more states in the interval dE). -In metals, the valence electrons fill up the states from the lowest energy with paired spins. For Na (sodium), each atom contributes one 3s electron and the electrons occupy all the states until fully used up. •The highest occupied level is called the Fermi level (EF). Density of states can be experimentally determined using X-ray emission spectroscopy. •High energy X-rays can remove core electrons. •The core energy levels are essentially atomic energy levels and have a discrete well-defined energy level. •Electrons from the conduction band can now fall to the energy level, emitting an X-ray. •The X-ray energy depends on the level of the conduction band of the falling electron. •A scan across the emitted X-rays will correspond to scan across the filled levels. •The intensity depends on the number of electrons with that particular energy. X-ray emission spectra X-ray emission spectra 3 Density of States for non-simple metals is more complex. Density of states may reach a maximum and then start to decrease with energy for more complex metals. •Extensions can be made to the model by including atomic nuclei and core electrons using a potential function (plane wave) to extend to semiconductors and insulators. •Limitations of the free electron model Electrical Conductivity Wave vectors, k, is important in understanding electrical conductivity in metals. •k is a vector with direction as well as magnitude. •There may be many different energy levels with the same value of k (same E), but different kx, ky, and kz and thus a different direction to the momentum. •In the absence of an electric field, all directions of the wave vector k are equally likely, so equal number of electrons are moving in all directions. 4 •If an electric field is produced (battery), then an electron traveling in the direction of the field will be accelerated and the energies of those levels with a net momentum in this direction is lowered. •Electrons moving in the opposite direction will have the energies raised and some of those electrons will fall into the levels of lower energy corresponding to momentum in the opposite direction. •More electrons will be moving in the direction of the field than in other directions, which is the electric current. *Empty levels close to the Fermi level must be available. Metals have finite resistance, governed by Ohm’s Law (V = iR) •Resistance of metals increases with temperature – for a given field the current decreases as temperature is raised. •To account for electrical resistance, it is necessary to consider the ionic cores. •Most crystals contain some imperfections and these can scatter the electrons. •This reduces the electron’s momentum and the current drops. •Even in perfect crystals, above 0 K the ionic cores will be vibrating, a set which is known as phonons. •Like vibrations in a molecule, each crystal vibration has a set of quantized energy vibrational levels. •Conduction electrons are scattered by these phonons and lose some energy to the phonons, lowering conductivity. •The energy loss to the phonons increase the vibrational energy of the crystal, converting electrical energy into thermal energy. •In some cases, this is useful for electrical heating. 5 As the temperature Lattice at high Lattice at low rises the atoms vibrate, temperature temperature acting as though they are larger. The conduction electrons are scattered by the vibrating ionic cores and lose some energy to the phonons, reducing electron flow and increasing the crystal’s vibrational energy. The net effect is to convert electrical energy to thermal energy. Conductor Conductivity Semiconductor Temperature 6 p Orbitals s Orbitals d Orbitals degenerate nondegenerate 7 Sigma () Bonds Hybrid Orbitals …four degenerate sp3 orbitals. 8 Molecular Orbital (MO) Theory Bond Order 1 1 (2 - 0) = 1 (2 - 2) = 0 2 2 9 • For atoms with both s and p orbitals, there are two types of interactions: – The s and the p orbitals that face each other overlap in fashion. – The other two sets of p orbitals overlap in p fashion. • The resulting MO diagram looks like this. • There are both and p bonding molecular orbitals and * and p* antibonding molecular orbitals. 10 • The smaller p-block elements in the second period have a sizeable interaction between the s and p orbitals. • This flips the order of the s and p molecular orbitals in these elements. 11 Molecular Orbital Theory Not all solids conduct electricity (insulators, semiconductors), so the free electron model is not a valid description for all solids. •Molecular Orbital theory treats all solids as a very large collection of atoms bonded together and try to solve the Schrödinger equation for a periodically repeating system. •The difficulty is that exact solutions have not yet been found for even small molecules, and certainly not for 1016 atoms. •One approximation that may be used in the linear combination of atomic orbitals (LCAO). H2 Molecule Chain of 5 H atoms c0 c1 c2 c3 c4 E Antibonding 4 c1 c2 3 Nonbonding 2 1 Bonding 0 # of Nodes 12 For N hydrogen atoms there will be N molecular orbitals. In between, there are (N-2) molecular orbitals with some in-phase and some out-of-phase combinations. As the number of atoms increases, the number of levels increases, but the spread of energies increase slowly and is leveling off for long chains. A metal crystal contains on the order of 1016 atoms and a range of energies of 10-19 J. The average separation between layers would be approximately 10-35 J. Compared to the energy separation of energy levels in a hydrogen atom (10-18 J), these separations are so small they may be thought of as a continuous range of energies know as an energy band. •It isn’t feasible to include all 1016 atoms in a calculation, so the periodicity of the crystal is used. The electron density and wave function for each unit cell is identical, so we form combinations of orbitals for the unit cell that reflect the periodicity of the crystal.
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